Find y(x) and f(y) given y''(x)=f(y) and int(f(y)dy)=x-1

[Bill_Nye_Fan]

Homework Statement

A function of y, $f(y)$, is known to be equal to the second derivative of function $y(x)$. ( i.e. $\frac{d^2y}{dx^2}=f\left(y\right)$ )

Given that $\int _{ }^{ }f\left(y\right)dy=x-1$, and function $y(x)$ has a stationary point at $x=1$ and an x-intercept at $x=2$, find $y(x)$ and hence find $f(y)$

Homework Equations

Knowledge of solving differential equations.

The Attempt at a Solution

$\int _{ }^{ }f\left(y\right)dy=x-1$
$\frac{d^2y}{dx^2}=f\left(y\right)$
$∴ \int _{ }^{ }\frac{d^2y}{dx^2}dy=\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy=x-1$
I derived both sides with respect to $y$
$\frac{d\left(\frac{dy}{dx}\right)}{dx}=\frac{d\left(x-1\right)}{dy}$
And this is where I get stuck. I'm not sure what to do from here, can someone please help? Thank you.

 

[blue_leaf77]

In
$$
\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy = \int \frac{dy}{dx} d\left(\frac{dy}{dx}\right)
$$
if you write $dy/dx = v$, can you do the integral analytically?

 

[Bill_Nye_Fan]

In
$$
\int _{ }^{ }\frac{d\left(\frac{dy}{dx}\right)}{dx}dy = \int \frac{dy}{dx} d\left(\frac{dy}{dx}\right)
$$
if you write $dy/dx = v$, can you do the integral analytically?

Ah yes I see what you're saying. Integrating a function (or in this case a derivative function) with respect to itself is the same as integrating a singular variable with respect to itself. I've done some quick working out and this is what I got:

Do you think this looks right or am I way off track?

 

[blue_leaf77]

If you check the consistency of your final result with the properties it should have as given in the problem, I guess then your answer is right.